Proof. Fix an orthonormal basis of \(H\)  and \(K\) , respectively.
Assume that \(W_\Phi \)  is optimal, by [1]}, \(W_\Phi -D\)  is not an
entanglement witness for any nonzero \(D\ge 0\) . Take any \(C\in {\mathcal {B}}(H,K)\)  and consider the map \(A\mapsto \Phi (A)-CAC^\dag \) .
Since the map \(A\mapsto CAC^\dag \)  is completely positive, the
corresponding Choi-Jamiołkowski matrix \(W_C\)  is positive. The
optimality of \(W_\Phi \)  implies that \(W_\Phi -W_C\)  is not an EW, and
so the map \(A\mapsto \Phi (A)-CAC^\dag \)  is not positive.
